The 1st day of creation, on which G-d created existence, time, matter, darkness and light, was the 25th of Elul. (Rosh Hashanah, on which we mark "the beginning of Your works", is actually the 6th day of creation, on which the world attained the potential for the realization of its purpose, with the creation of the first man and woman, Adam and Eve. Rosh Hashanah is therefore the day from which the Jewish calendar begins to count the years of history; the 1st day of creation thus occurred on the 25th of Elul of what is termed -1 from creation.
The rebuilding of the walls of Jerusalem -- which had been in ruins since the destruction of the First Temple by the Babylonians 88 years earlier -- was completed by Nehemia on Elul 25 of the year 3426 from creation (335 BCE) as related in the Book of Nehemia (ch. 6).
Passing of the Talmudic sage Rabbi Elazar, son of Rabbi Shimon bar Yochai.
Passing of Rabbi Yechiel Michel of Zlotchov (1721?-1786), disciple of the Baal Shem Tov.
The Selichot ("supplication") prayers are recited in the early morning hours, before the morning prayers, in preparation for the "Days of Awe" of Rosh Hashanah and Yom Kippur.
Links: More on Selichot
As the last month of the Jewish year, Elul is traditionally a time of introspection and stocktaking -- a time to review one's deeds and spiritual progress over the past year and prepare for the upcoming "Days of Awe" of Rosh HaShanah and Yom Kippur.
As the month of Divine Mercy and Forgiveness (see "Today in Jewish History" for Elul 1) it is a most opportune time for teshuvah ("return" to G-d), prayer, charity, and increased Ahavat Yisrael (love for a fellow Jew) in the quest for self-improvement and coming closer to G-d. Chassidic master Rabbi Schneur Zalman of Liadi likens the month of Elul to a time when "the king is in the field" and, in contrast to when he is in the royal palace, "everyone who so desires is permitted to meet him, and he receives them all with a cheerful countenance and shows a smiling face to them all."
Specific Elul customs include the daily sounding of the shofar (ram's horn) as a call to repentance. The Baal Shem Tov instituted the custom of reciting three additional chapters of Psalms each day, from the 1st of Elul until Yom Kippur (on Yom Kippur the remaining 36 chapters are recited, thereby completing the entire book of Psalms). Click below to view today's Psalms.
Chapter 73
Chapter 74
Chapter 75
Elul is also the time to have one's tefillin and mezuzot checked by an accredited scribe to ensure that they are in good condition and fit for use.
Links: More on Elul